# What is the difference between Boolean algebra and statement logic?

How do you craft a statement logic?

First of all, we are looking for a nice association with many good characteristics[math(L,’land,’lor,’lnot)[/math i.e. with all the rule of a **Bool’s association or**Boolean algebra (see Wikipaedia) where L represents a lot of statements, [math’land[/math is equal to AND ,[math’lor [/math is equal to OR and [math’lnot[/math equals NOT.In the Enlische (AND, OR, NOT).

A statement is naively something to which I have a truth value (TRUE or FALSE) or(1,0), or .1, 1/2, 0, etc.

Now we need a second association in which we want to count.

[math(F,’land,’lor,’lnot) [/mathwith an ordered quantity F but otherwise all the estates of the first association, in which we want to reckon with the truth values of the statements.For a two-valued logic usually F := [math-TRUE,FALSE-[/math

in mathematics, one takes

F := [math——————————————————————————————————————————————————————————————————————————————————————————— /math We want to calculate.

How is the calculation now?With x,y [math-epsilon[/math F

How are ([math-land,-lor,-lnot[/math) defined in the second bandage?

- [mathx sland y[/math := [mathmin(x,y)[/math
- [mathx slor y [/math := [mathmax(x,y)[/math
- [math-lnot x[/math := [math1 – x[/math

(F,min,max, [math-lnot)[/math is a Galois body

Thirdly, we need a truth value function [math-omega[/math, which represents an association homomorphism and mediates between the two associations in an operational manner.In doing so, the statements are assigned to logical values, with which we can then “calculate” the true value of logical formula

[math-omega : L-rightarrow F,[/math with p,q [math-epsilon [/math L

- [math-omega(p-land q)[/math := [math-omega(p)
- [math-omega(p-lor q)[/math := [math-omega(p)

[math-omega(-lnot p)[/math := [math-lnot-omega(p[/math) := [math1 – .omega(p)[/math

- This gives us a
**logic of statements**

**Example:**

p :=” The moon is made of cheese”

q := “2 +2 = 4”

[math-omega(p) := 0[/math

[math-omega(q) := 1[/math

[math-omega(-lnot(p-land q)) = .lnot-omega(p-land q) [/math

[math= slnot(-omega(p) -country————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————

[math=-lnot (min(0,1)) = .lnot( 0) = 1 – 0 = 1[/math,

so the AND-linking of both pronouncements is FALSE and the negation of this is TRUE again

**Voila!**

**Attention; complementary** [math-omega(p) = 1 .implies -lnot -omega(p) = 0 [/math

[math-implies [/math

[math-omega(p-land-lnot(p)) [/math

[math= somega(p) s.o.o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and analog

[math-omega(p) = 1 .implies -lnot -omega(p) = 0 [/math

[math-implies [/math

[math-omega(p-lor-lnot(p)) [/math

[math= .omega(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[math= max(1,0) = 1[/math

For a **discrete trivalent logic according to the following eukasiewicz**, choose

[mathF := .1, 1/2, 0. [/math

The operations in F, as above, leave

**Example:**

p := ” The sun shines”

q := “Peter comes to swim”

[math-omega(p)[/math := 1

[math-omega(q)[/math := 1/2 (= DETO)

[math-omega(p-land q) = .omega(p) – country ———————————————————————————————————————————————————–

**Attention; non-complementary** [math_implies[/math ** non-boolsch**[math -omega(p) = 1/2 .implies-lnot-omega(p) [/math

[math= 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[math= .omega(p) – land – omega(-lnot(p)) [/math

[math= min(1/2,1/2) =1/2

- For L ;= [0,1 we have a
**constant multi-value logic**

**Attention: non-complementary** [math-implies[/math ** not boolsch** s. Top

- For[math F_[a,b’ := .f:[a,b-rightarrow [0,1|a,b -epsilon-r, a < b-[/math we have a
**fuzzy logic**, with

[math-omega : L-rightarrow F_[a,b-[/math

[math p-longmapsto f [/math here functions are assigned to the statements as truth values

the operations are defined by point.

Be[math f,g sepsilon F_[a,b-[math and x [math-epsilon[/math [a,b

- [math(f-land g)(x) = f(x) -land g(x) := min(f(x),g(x))[/math
- [math(f-lor g)(x) = f(x) -lor g(x) := max(f(x),g(x))[/math
- [math(lnot f)(x) = slnot f(x) := const_1(x) – f(x)[/math

**Attention: non-complementary** [math_implies[/math ** not boolsch**

in general [math(f’land’lnot f)(x) = min (f(x), const_1(x) – f(x))

**Example: s.**